Theorem 3.19 (0N48) · ScalingStacks

0N48

Theorem 3.19. For an associative algebra $A$ in a symmetric monoidal $\oo$ -category $\mathcal{V}$ which is $\otimes$ -presentable, there is an equivalence

\int_{S^{1}}A~\simeq~\hh_{*}(A)

between the factorization homology of the circle with coefficients in $A$ and the Hochschild complex of $A$ .

0N49

Proof. Regard the associative algebra $A$ as a symmetric monoidal functor $A\colon\disk^{\sf or}_{1}\to\mathcal{V}$ , as in Section 3.2. Consider the standard collar-gluing $\mathbb{R}\underset{\mathbb{R}\sqcup\mathbb{R}}{\bigcup}\mathbb{R}\cong S^{1}$ by hemispheres. Lemma 3.18, which states that factorization homology staisfies $\otimes$ -excision, determines the first of the equivalences in the expression:

\int_{S^{1}}A~\simeq~\int_{\mathbb{R}}A\underset{{\displaystyle\int_{{S^{0}\times\mathbb{R}}}\!A}}{\bigotimes}\int_{\mathbb{R}}A~\simeq~A\underset{A\otimes A^{\op}}{\bigotimes}A~\simeq~\hh_{*}(A)~.

The second equivalence is by inspecting values, and the final equivalence is definitional.

∎